A PARAMETER-UNIFORM TAILORED FINITE POINT METHOD FORSINGULARLY PERTURBED LINEAR ODE SYSTEMS

Houde HanDepartment of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Email: hhan@math.tsinghua.edu.cn.J. J. H. Miller

Department of Mathematics, Trinity College, Dublin 2, IrelandEmail: jmiller@tcd.ie.

Min TangKey Laboratory of Scientific and Engineering Computing,

Department of Mathematics and Institute of Natural Sciences, Shanghai Jiao Tong University,Shanghai, China, 200240.

Email: tangmin@sjtu.edu.cn

Abstract

In scientific applications from plasma to chemical kinetics, a wide range of temporal scales can presentin a system of differential equations. A major difficulty is encountered due to the stiffness of the system andit is required to develop fast numerical schemes that are able to access previously unattainable parameterregimes

We consider an initial-final value problem for a multi-scale singularly perturbed system of linear ordinarydifferential equations with discontinuous coefficients. We construct a tailored finite point method, whichyields approximate solutions that converge in the maximum norm, uniformly with respect to the singularperturbation parameters, to the exact solution. A parameter-uniform error estimate in the maximum normis also proved. The results of numerical experiments, that support the theoretical results, are reported.

Mathematics subject classification: 37M05, 65G99Key words: Tailored finite point method, parameter uniform, singular perturbation, ODE system

1. Introduction

We consider the following initial-final value problem for a system of linear ordinary differential equations withdiscontinuous coefficients

Eu′(t) +A(t)u(t) = f(t), ∀t ∈ (pk, pk+1), k = 0, . . . ,K, (1.1)

u(pk + 0)− u(pk − 0) = 0, k = 1, . . . ,K, (1.2)

Bεu(0) + (I− Bε)u(1) = d, (1.3)

where E , Bε are n× n matrices, d is a vector, A(t) is an n× n matrix function and f(t) is a vector function onthe interval [0, 1] such that

A(t) =(ai,j(t)

)n×n, (1.4)

f(t) = (f1(t), f2(t), . . . , fn(t))T (1.5)

and {pk}K+10 are some numbers which satisfy

0 = p0 < p1 < · · · < pK < pK+1 = 1.

The given functions ai,j(t), fi(t) (1 ≤ i, j ≤ n) may not be continuous on the whole interval [0, 1]. Here,we consider the case when they are piecewise continuous. More precisely, we assume that the functions

http://www.global-sci.org/jcm Global Science Preprint

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ai,j(t), fi(t) (1 ≤ i, j ≤ n) have K points of discontinuity of the first kind, t = pk, (0 < pk < 1; k = 1, . . . ,K), sothat on each subinterval (pk, pk+1), (k = 0, . . . ,K; p0 = 0, pK+1 = 1) the functions are smooth and satisfy theconditions

ai,i(t)−n∑

j=1,j 6=i

|ai,j(t)| ≥ β > 0, ∀t ∈ [0, 1]. (1.6)

Furthermore, we assume that the matrices Eε = diag(ε1, ε2, . . . , εn) and Bε = diag(b1, b2, . . . , bn) are diagonaland satisfy the conditions

|εi| > 0, 1 ≤ i ≤ n (1.7)

and

bi =

{1, εi > 0,

0, εi < 0.(1.8)

We also suppose that there exists at least one εj (1 ≤ j ≤ n) such that

0 < |εj | � 1. (1.9)

Problem (1.1)-(1.3) is then an initial-final value problem for a multi-scale singularly perturbed system of linearordinary differential equations with discontinuous coefficients. The solution of problem (1.1)-(1.3) may containinitial, final and interior layers at any of the points pk (k = 1, . . . ,K). The main goal in this paper is to developa class of numerical methods, which yield approximate solutions that converge in the maximum norm, uniformlywith respect to the singular perturbation parameters, to the exact solution of this problem.

When all of the parameters (εj , j = 1, . . . , n) are positive, problem (1.1)-(1.3) reduces to the initial valuesingularly perturbed problem

Lu(t) ≡ Eu′(t) +A(t)u(t) = f(t), ∀t ∈ (pk, pk+1), k = 0, . . . ,K, (1.10)

u(pk + 0)− u(pk − 0) = 0, k = 1, . . . ,K, (1.11)

u(0) = d, (1.12)

which has been studied in [23].They proposed a Shishkin piecewise uniform mesh with a classical finite differencescheme to obtain numerical solutions of this problem; a parameter-uniform error estimate was also given.

To motivate the study of the more general initial-final value problem in the paper, it should be noted that asemi-discretization, with respect to variable x, of the following forward-backward parabolic problem

sign(x)|x|p ∂u(x, t)

∂t=

∂2u(x, t)

∂x2− σ(x)u(x) + q(x, t), −1 < x < 2, 0 < t < T, (1.13)

u∣∣x=−1

= f(t), u∣∣x=2

= g(t), 0 < t < T, (1.14)

u∣∣t=0

= s(x), 0 < x < 2, u∣∣t=1

= γ(x), −1 < x < 0. , (1.15)

with p > 1, σ(x) > 0 can lead to the initial-final singularly perturbed problem (1.1)-(1.3). This is because|x|p ∈ (0, 2p) ranges from very small to 2p.

In this paper we construct a parameter-uniform scheme for the initial-final multi-scale singularly perturbedproblem(1.1)-(1.3) using the tailored finite point method (TFPM). The TFPM was proposed by Han, Huangand Kellogg in [9] for the numerical solution of singular perturbation problems. The basic idea of the TFPM is

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to choose, at each mesh point, suitable basis functions based on the local properties of the solutions of the givenproblem; then to approximate the solution using these basis functions. At each point, the numerical schemeis tailored to the given problem. The TFPM was successfully applied by Han, Huang ,and Kellogg to solvethe Hemker problem [9, 11]; they won the Hemker prize at the international conference BAIL 2008. Later theTFPM was developed to solve the second-order elliptic singular pertubation problem [6, 21], the first order waveequation [13], the one-dimension Helmholtz equation with high wave number [5], second order elliptic equationswith rough or highly oscillatory coefficients [10] and so on [7, 8]. For the one dimensional singular perturbationproblem the TFPM is close to the method of “exponential fitting” discussed in [1, 3, 16, 20]. The TFPM is alsoapplied to the one-dimensional discrete-ordinate transport equations in [17].

The TFPM uses the functions that exactly satisfy the PDE as the bases. For the linear ordinary differentialequations system under consideration, the bases are exponential functions. The exponential integrator appearsfor a long time, see the review paper [19]. The new idea in TFPM is to approximate all the coefficients A(t)and f(t) by piecewise contants. Compared with previous magnus integrators [12] and adiabatic integrators [18]developed for the stiff problem or highly oscillatory problems, our approach is simple and proved to possessparameter uniform convergence. More precisely, we can use time steps much larger than the parameters εi in(1.7) and achieve stable and accurate results.

The main contribution of this paper is that we construct a tailored finite point method for (1.1)-(1.3), whichyields approximate solutions that converge in the maximum norm, uniformly with respect to the singularperturbation parameters, to the exact solution. The parameter-uniform error estimate in the maximum norm isproved analytically as stated in Theorem 4.1. Some numerical experiments that support the theoretical resultsare presented in section 5.

2. Existence and uniqueness of solutions to the problem (1.1)-(1.3)

In this section we discuss the existence and uniqueness of solutions to the problem (1.1)-(1.3). On [0, 1] weintroduce the following function spaces

C(0)∗,∗[0, 1] = {v(t)

∣∣v|(pk,pk+1) ∈ C(0)(pk, pk+1), k = 0, 1, . . . ,K}, (2.1)

C(1)∗ [0, 1], =

{v(t)

∣∣v(t) ∈ C(0)[0, 1], and v′(t) ∈ C(0)∗∗ [0, 1]

}, (2.2)

C(1)∗,∗[0, 1] =

{v(t)

∣∣v(t), v′(t) ∈ C(0)∗∗ [0, 1]

}. (2.3)

Note that the space C(0)[0, 1] is a subspace of C(0)∗∗ [0, 1] and that, for any v(t) ∈ C(0)

∗,∗[0, 1], v(t) may havediscontinuity points of the first kind t = pk (k = 1, . . . ,K). On each subinterval [pk, pk+1] we adopt the

conventions that v(pk) = limt→p+kv(pk), v(pk+1) = limt∈p−k+1

v(pk+1). The space C(1)∗ [0, 1] is a subspace of

C(1)∗,∗[0, 1].

For any v(t) ∈ C(0)∗,∗[0, 1], we define the norm

‖v‖∞ = max0≤k≤K

{max

pk≤t≤pk+1

|v(t)|}

(2.4)

and for any v(t) ∈ C(1)∗,∗[0, 1], we define the norms

|v|1,∞ = max0≤k≤,K

{max

pk≤t≤pk+1

|v′(t)|}, (2.5)

‖v‖1,∞ = max{‖v‖∞, |v|1,∞

}. (2.6)

Furthermore, we introduce the following function vector and matrix spaces

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C(1)∗ [0, 1] =

(C(1)∗ [0, 1]

)n, (2.7)

C(1)∗,∗[0, 1] =

(C(1)∗,∗[0, 1]

)n, (2.8)

C(1)∗,∗[0, 1] =

(C(1)∗,∗[0, 1]

)n×n. (2.9)

Note that C(1)∗ [0, 1] is a subspace of C(1)

∗,∗[0, 1]. For any v(t) ∈ C(1)∗,∗[0, 1], we define the norms ‖·‖∞, |·|1,∞, ‖·‖1,∞

‖v‖∞ = max0≤k≤K

(max

pk≤t≤Pk+1

‖v(t)‖∞), (2.10)

|v|1,∞ = ‖v′‖∞, (2.11)

‖v‖1,∞ = max(‖v‖∞, ‖v′‖∞

), (2.12)

where the norm ‖v(t)‖∞ is the norm in the vector space Rn for fixed t ∈ [0, 1]. Similarly we define the norms

‖ · ‖∞, | · |1,∞, ‖ · ‖1,∞ in the space C(1)∗,∗[0, 1].

For problem (1.1)-(1.3), we then have the following theorem.

Theorem 2.1. Suppose that A(t) ∈ C(0)∗,∗[0, 1], f(t) ∈ C(0)

∗,∗[0, 1] and A(t), f(t) satisfy the conditions (1.4)-(1.9).

Then, for problem (1.1)-(1.3) there exists a unique solution uε(t), uε(t) ∈ C(1)∗ [0, 1], and the following estimate

holds

‖uε‖∞ ≤ max{‖d‖∞,

‖f‖∞β

}. (2.13)

Proof. We first establish the estimate (2.13). Suppose that uε(t) = (uε1(t), . . . , uεn(t))T ∈ C(1)∗ [0, 1] is a

solution of problem (1.1)-(1.3). Let

M = ‖uε‖∞. (2.14)

By the continuity of uε(t) on the interval [0, 1], we can find a point tm ∈ [0, 1], such that

M = |uεi (tm)|,

for some integer i , 1 ≤ i ≤ n. If M = 0, the estimate follows directly.Otherwise, we consider the case uεi (tm) > 0 ( for the case uεi (tm) < 0, the proof is similar ). Then we know that

|uεj(t)| ≤ uεi (tm) = M, ∀t ∈ [0, 1], 1 ≤ j ≤ n. (2.15)

(i) If tm 6= pk, k = 0, 1, . . . ,K + 1, the inequality (2.15) yields

uεi′(tm) = 0. (2.16)

From system (1.1), we obtain

ai,i(tm)uεi (tm) +∑

1≤j≤n,j 6=i

ai,j(tm)uεj(tm) = fi(tm) (2.17)

and the estimate follows immediately from (1.6).(ii) If tm = pk, k = 0, 1, . . . ,K + 1, suppose that εi > 0 ( for the case εi < 0 , the proof is similar ), If k = 0,then tm = 0, and so, from the initial- final condition (1.3 ), we know that M = uεi (tm) = di, which yields the

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estimate (2.13). Otherwise, if 0 < k ≤ K + 1, note that, on the interval [pk−1, pk], the function uεi (t) satisfiessystem (1.1) at the the point t = pk, and so

ai,i(tm − 0)uεi (tm) +∑

1≤j≤n,j 6=i

ai,j(tm − 0)uεj(tm) ≤ fj(tm − 0). (2.18)

This completes the proof of the estimate (2.13).Using estimate (2.13), we can now show that Problem (1.1)-(1.3) has a unique solution. To see this we note

that the following homogeneous initial-final value problem

Eu′(t) +A(t)u(t) = 0, ∀t ∈ (pk, pk+1), k = 0, . . . ,K (2.19)[u(pk + 0)− u(pk − 0)

]= 0, k = 1, . . . ,K, (2.20)

Bεu(0) + (I− Bε)u(1) = 0, (2.21)

has a unique solution uε(t) = 0.To prove the existence of a solution to problem (1.1)-(1.3), we introduce the following auxiliary initial value

problems

Eu′(t) +A(t)u(t) = f(t), ∀t ∈ (pk, pk+1), k = 0, . . . ,K (2.22)

u(pk + 0)− u(pk − 0) = 0, k = 1, . . . ,K, (2.23)

u(0) = 0 (2.24)

and

Eu′(t) +A(t)u(t) = 0, ∀t ∈ (pk, pk+1), k = 0, . . . ,K (2.25)

u(pk + 0)− u(pk − 0) = 0, k = 1, . . . ,K, (2.26)

u(0) = ej , (2.27)

for each integer j (1 ≤ j ≤ n).By the existence theorem for the initial value problem for the linear ODE system, we know that for problem

(2.22)-(2.24) there exists a unique solution vεf (t) = (vε1,f (t), . . . , vεn,f (t))T ∈ C(1)∗ [0, 1] , and for each integer

1 ≤ j ≤ n, for problem (2.25)-(2.27) there exists a unique solution vεj(t) = (vεj,1(t), . . . , vεj,n(t))T ∈ C(1)∗ [0, 1] .

The general solutions of system (1.1) are given by

uε(t) =∑

j=1,...,n

cjvεj(t) + vεf (t), (2.28)

for arbitrary constants cj , (j = 1, . . . , n).Let c = (c1, . . . , cn)T and

Vε(t) = (vε1(t), . . . ,vεn(t)), (2.29)

where Vε(t) is an n× n matrix function. Then equality (2.28) can be rewritten as follows

uε(t) = Vε(t)c + vεf (t). (2.30)

The vector-valued function uε(t) given by (2.30) satisfies system (1.1) and the continuity condition (1.2) forany constant vector c. Thus, if we can find a vector c ∈ Rn, such that uε(t) satisfies the initial-final condition(1.3), the existence of a solution to problem (1.1)-(1.3) has been proved. The initial-final condition (1.3) yields

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[BεVε(0) + (I− Bε)Vε(1)

]c = d−

[Bεvεf (0) + (I− Bε)Vε

f (1)]. (2.31)

The uniqueness of the solution of problem (1.1)-(1.3) leads to

Det∣∣∣BεVε(0) + (I− Bε)Vε(1)

∣∣∣ 6= 0. (2.32)

This implies that for system (2.31) there exists a unique solution c ∈ Rn. Then, from (2.30), we obtain asolution of problem (1.1)-(1.3).

This completes the proof.

3. An approximation to problem (1.1)-(1.3)

In this section we construct an approximation to problem (1.1)-(1.3). The interval [0, 1] is divided intosubintervals by

0 = t0 < t1 < · · · < tL = 1, (3.1)

such that, for each pk ∈ [0, 1], we can find a tl such that pk = tl. Furthermore, let

hl = tl − tl−1,∀ l = 1, . . . , L; h = maxl=1,...,L

(hl). (3.2)

On each subinterval (tl−1, tl) the functions ai,j(t), fi(t) are approximated by the constants ai,j(t∗l ), fi(t

∗l ) with

tl−1 ≤ t∗l ≤ tl. Then we introduce the matrix and vector

Al =(ai,j(t

∗l ))n×n

, (3.3)

f l = (f1(t∗l ), f2(t∗l ), . . . , fn(t∗l ))T . (3.4)

We also define the approximating matrix and vector functions

Ah(t) = Al, ∀t ∈ (tl−1, tl), (3.5)

fh(t) = f l, ∀t ∈ (tl−1, tl). (3.6)

It is easy to see that for these approximating functions, the following estimates hold

‖fh‖∞ ≤ ‖f‖∞, (3.7)

‖f − fh‖∞ ≡ maxl=0,1,...,L

maxtl−1≤t≤tl

‖f(t)− fh(t)‖∞ ≤ ‖f ′‖∞h, (3.8)

‖A −Ah‖∞ ≡ maxl=0,1,...,L

maxtl−1≤t≤tl

‖A(t)−Ah(t)‖∞ ≤ c‖A′‖∞h, (3.9)

where c is a constant, independent of h.Now consider the approximate problem

Euεh′(t) +Ah(t)uεh(t) = fh(t), ∀t ∈ (tl−1, tl), l = 1, . . . ,K, (3.10)

uεh(tl + 0)− uεh(tl − 0) = 0, l = 1, . . . , L− 1, (3.11)

Bεuεh(0) + (I− Bε)uεh(1) = d. (3.12)

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By Theorem 2.1, we know that for problem (3.10)-(3.12) there exists a unique solution uεh(t), and that thefollowing estimate holds

‖uεh‖∞ ≤ max{‖d‖∞,

‖f‖∞β

}. (3.13)

We introduce the error function rεh(t) = uε(t)− uεh(t). Then rεh(t) satisfies

Erεh′(t) +Ah(t)rεh(t) = dεh(t), ∀t ∈ (tl−1, tl), l = 1, . . . ,K, (3.14)

rεh(tl + 0)− rεh(tl − 0) = 0, l = 1, . . . , L− 1, (3.15)

Bεrεh(0) + (I− Bε) rεh(1) = 0, (3.16)

with

dεh(t) = f(t)− fh(t) +(Ah(t)−A(t)

)uεh(t). (3.17)

Combining equality (3.17) and inequalities (3.7)-(3.9),(3.13), we arrive at

‖dεh‖∞ ≤{‖f ′‖∞ + c‖A′‖∞

(‖d‖∞ +

‖f‖∞β

)}h, (3.18)

where c is a constant independent of h and ε.Applying Theorem 2.1 to problem (3.14)-(3.16), we see that the following error bound holds

Theorem 3.1. The error rεh satisfies the estimate

‖rεh‖∞ ≤1

β

{‖f ′‖∞ + c‖A′‖∞

(‖d‖∞ +

‖f‖∞β

)}h, (3.19)

where c is a constant independent of h and E .This shows that the approximate solution uεh(t) converges E-uniformly to the solution uε(t) of problem (1.1)-

(1.3) on the interval [0, 1].

4. A tailored finite point method for problem (1.1)–(1.3)

Using uεh(t) for the solution of Problem (3.9)–(3.11), we now introduce the tailored finite point method forproblem (1.1)–(1.3). Let

uεh(tl) = ul, l = 0, 1, . . . , L. (4.1)

Then on each subinterval [tl−1, tl],the approximate solution uεh(t) satisfies

Euεh′(t) +Aluεh(t) = f l, ∀t ∈ (tl−1, tl), (4.2)

Bεuεh(tl−1) + (I− Bε)uεh(tl) = Bεul−1 + (I− Bε)ul. (4.3)

It is easy to see that the ODE system (4.2) with constant coefficients has the particular solution

vlf =(Al)−1

f l. (4.4)

We now construct the general solution vεh of the homogeneous ODE system corresponding to (4.2)

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Evεh′(t) +Alvεh(t) = 0, ∀t ∈ (tl−1, tl). (4.5)

Let

vεh(t) = ξeλt (4.6)

be the solution of system (4.5). Then the vector ξ and number λ are the solutions of the eigenvalue problem

Alξ = −λEξ. (4.7)

Solving the eigenvalue problem (4.7), we obtain the eigenvalues λl1, λl2, . . . , λ

l2r, λ

l2r+1, λ

ln, corresponding to

the eigenvectors ξl1, ξl2, . . . , ξ

l2r, ξ

l2r+1, . . . , ξ

ln. Assume that the eigenvalues λl2r+1, . . . , λ

ln and the corresponding

eigenvectors are real. The remaining eigenvalues occur in complex conjugate pairs, so that

λl2j = λl2j−1, j = 1, . . . , r, (4.8)

ξl2j = ξl2j−1, j = 1, . . . , r, (4.9)

These conjugate pairs of eigenvalues λl2j−1, λl2j and eigenvectors ξl2j−1, ξ

l2j , can be written in the form

λl2j−1 = αlj − iβlj ,λl2j = αlj + iβlj ,

ξl2j−1 = ηlj − iζlj ,ξl2j = ηlj + iζlj .

Then, for each 1 ≤ j ≤ r, there are two independent solutions of the system (4.5) of the form

vl2j−1(t) =(ηlj cos(βljt)− ζlj sin(βljt)

)eα

lj(t−tl), (4.10)

vl2j(t) =(ηlj cos(βljt) + ζlj sin(βljt)

)eα

lj(t−tl). (4.11)

Furthermore, for each j (2r < j ≤ n), there is one solution of the system (4.5) given by

vlj(t) = ξljeλlj(t−tl). (4.12)

Let

V l(t) ≡(vl1(t),vl2(t), . . . ,vln(t)

), (4.13)

where V l(t) is a n× n matrix function. On each interval [tl−1, tl], uεh(t) is given by

uεh(t) = V l(t)cl + vlf , (4.14)

with cl = (cl1, . . . , cln)T , which is determined by the initial-final condition (4.3). Then we have

cl =(BεV l(tl−1) +

(I− Bε

)V l(tl)

)−1(Bεul−1 +

(I− Bε

)ul − vlf

)≡ Dl

(Bεul−1 +

(I− Bε

)ul − vlf

), (4.15)

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with

Dl =(BεV l(tl−1) +

(I− Bε

)V l(tl)

)−1

.

Furthermore, on each interval [tl−1, tl], the expression (4.14) can be written as

uεh(t) = V l(t)Dl(Bεul−1 +

(I− Bε

)ul − vlf

)+ vlf . (4.16)

The continuity condition (3.12) yields

(I− Bε

)ul−1 =

(I− Bε

){V l(tl−1)Dl

(Bεul−1 +

(I− Bε

)ul − vlf

)+ vlf

}, (4.17)

Bεul = Bε{V l(tl)Dl

(Bεul−1 +

(I− Bε

)ul − vlf

)+ vlf

}. (4.18)

We then obtain the following tailored finite point scheme

ul =(I− Bε

){V l+1(tl)Dl+1

(Bεul +

(I− Bε

)ul+1 − vl+1

f

)+ vl+1

f

}(4.19)

+ Bε{V l(tl)Dl

(Bεul−1 +

(I− Bε

)ul − vlf

)+ vlf

}, l = 1, 2, . . . , L− 1,

u0 = Bεd +(I− Bε

){V1(t0)D1

(Bεd +

(I− Bε

)u1 − v1

f

)+ v1

f

}, (4.20)

uL =(I− Bε

)d + Bε

{VL(tL)DL

(BεuL−1 +

(I− Bε

)d− vLf

)+ vLf

}. (4.21)

In the special case when the {εj , j = 1. . . . , n} are all positive, we have Bε = I and the method (4.19)-(4.21),for finding the numerical solution of problem (1.10)-(1.12), is reduced to the following method

ul = V l(tl)Dl(ul−1 − vlf

)+ vlf , l = 1, 2, . . . , L, (4.22)

u0 = d. (4.23)

This is a one step explicit scheme, which is unconditionally stable.

Since the discrete scheme (4.19)-(4.21) is equivalent to the approximate problem (3.9)-(3.11), we attain thefollowing theorem;

Theorem 4.1. Suppose that A(t) ∈ C(1)∗,∗[0, 1], f(t) ∈ C(1)

∗,∗[0, 1] and that A(t), f(t) satisfy the conditions (1.3)-(1.9). Then there exists a unique solution {ul l = 0, 1, . . . , L} to problem (4.19)-(4.21) and the followingparameter-uniform error estimate holds

maxl=0,1,...,L

‖u(tl)− ul‖∞ ≤1

β

{‖f ′‖∞ + c‖A′‖∞

(‖d‖∞ +

‖f‖∞β

)}h. (4.24)

This shows that the tailored finite point method yields approximate solutions that are parameter-uniformlyconvergent with respect to {0 < |εj | ≤ 1, ∀j = 1. . . . , n} to the solution of the multi-scale singularly perturbedproblem (1.1)-(1.3).

Solving the algebraic equation (4.19)-(4.21) gives the numerical solution {ul, l = 0, 1, . . . , L}. Combiningthis with the expressions (4.15), (4.14) then yields the required continuous approximate solution uεh(t).

10

5. Numerical examples

In this section three problems are solved numerically using a tailored finite point method. The first is aninitial value problem and the second an initial-final value problem for a singularly perturbed system of ordinarydifferential equations. The third problem arises from a semi-discretization of a forward-backward parabolicproblem. The errors in the numerical solutions for the first two problems, as well as numerical estimates of theparameter-uniform rates of convergence and the parameter-uniform error constants, are given in five tables. InTables 1, 3, 5 the errors of the numerical solutions are first calculated by comparing the numerical solutions withthe ”exact” solutions that are obtained with the finest mesh ∆t = 1/4096. On the other hand, in Tables 2, 4parameter-uniform estimates of the convergence rates and error constants are computed using the methodologydescribed in [4]. Graphs of the solutions of all three problems are presented in the figures.Example 1. For t ∈ (0, 1), we solve

ε1u′1(t) + 4u1(t)− u2(t)− u3(t) = t

ε2u′2(t)− u1(t) + 4u2(t)− u3(t) = 1

ε3u′3 − u1(t)− u2(t) + 4u3(t) = 1 + t2

u1(0) = 0, u2(0) = 0, u3(0) = 0.

for different values of ε1, ε2 and ε3. This example appears in [23] and we compare our tailored finite pointmethod with the classical finite difference discretization using a Shishkin piecewise uniform mesh. Both methodsare seen to be first order parameter-uniform. Similarly to [23], let ε1 = r/16, ε2 = r/4, ε3 = r. The numericalsolutions of u1(t) for different r are displayed in Figure 5.1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t

u1

r=1/128

r=1/32

r=1/8

r=1/2

Fig. 5.1. Example 1. u1(t) for r = 1/2, 1/8, 1/32, 1/128.

The L∞ norm of the numerical errors for different r and time step are given in Table 5.1, where we have usedt∗l = tl−1 in the formulas (3.3) and (3.4). Parameter-uniform first order convergence for all r can be observedin this table,

The parameter-uniform error parameters p∗ and C∗p∗ are given by the well-established two-mesh procedurefor numerically finding a parameter-uniform error bound of the form

‖U∆tε − uε‖∞ ≤ C∗p∗∆tp

∗,

11

r \ ∆t 1/128 1/256 1/512 1/1024 1/2048

2−1 2.311 ∗ 10−3 1.115 ∗ 10−3 5.194 ∗ 10−4 2.224 ∗ 10−4 7.412 ∗ 10−5

2−2 2.507 ∗ 10−3 1.205 ∗ 10−3 5.605 ∗ 10−4 2.398 ∗ 10−4 7.987 ∗ 10−5

2−3 2.638 ∗ 10−3 1.258 ∗ 10−3 5.829 ∗ 10−4 2.489 ∗ 10−4 8.283 ∗ 10−5

2−4 2.768 ∗ 10−3 1.300 ∗ 10−3 5.979 ∗ 10−4 2.544 ∗ 10−4 8.449 ∗ 10−5

2−5 2.967 ∗ 10−3 1.354 ∗ 10−3 6.132 ∗ 10−4 2.589 ∗ 10−4 8.568 ∗ 10−5

2−6 3.315 ∗ 10−3 1.448 ∗ 10−3 6.373 ∗ 10−4 2.649 ∗ 10−4 8.700 ∗ 10−5

2−7 3.877 ∗ 10−3 1.619 ∗ 10−3 6.820 ∗ 10−4 2.757 ∗ 10−4 8.917 ∗ 10−5

2−10 5.139 ∗ 10−3 2.418 ∗ 10−3 1.057 ∗ 10−3 4.010 ∗ 10−4 1.174 ∗ 10−4

2−15 5.280 ∗ 10−3 2.559 ∗ 10−3 1.195 ∗ 10−3 5.124 ∗ 10−4 1.708 ∗ 10−4

2−16 5.280 ∗ 10−3 2.559 ∗ 10−3 1.195 ∗ 10−3 5.124 ∗ 10−4 1.708 ∗ 10−4

2−17 5.280 ∗ 10−3 2.559 ∗ 10−3 1.195 ∗ 10−3 5.124 ∗ 10−4 1.708 ∗ 10−4

Table 5.1: Example 1. L∞ norm of the numerical errors, for different r and time steps, when t∗l = tl−1 in (3.3) and (3.4),ε1 = r/16, ε2 = r/4, ε3 = r

which is described in [4]. The technique uses the two-mesh method, which involves the quantities

D∆tε = ‖U∆t

ε − U∆t/2ε ‖∞, D∆t = max

ε{D∆t

ε },

p∆t = log2

D∆t

D∆t/2, p∗ = min

∆tp∆t

and

C∆tp∗ =

D∆t

∆tp∗(1− 2−p∗), C∗p∗ = max

∆tC∆tp∗

For this example the results are given in Table 2, where it is seen that p∗ = 0.996 and C∗p∗ = 0.685.

If we use

Al =1

tl − tl−1

(∫ tl

tl−1

ai,j(t) dt)n×n

, (5.1)

f l =1

tl − tl−1(

∫ tl

tl−1

f1(t) dt,

∫ tl

tl−1

f2(t) dt, . . . ,

∫ tl

tl−1

fn(t) dt)T , (5.2)

instead of t∗l = tl−1, the corresponding errors are displayed in Table 5.3. It can be seen that parameter-uniform first order convergence is achieved. However, second order convergence occurs when r is large, whilewhen r decreases, only first order convergence is attained. This is different from the one dimensional neutrontransport equation, where with cell averaging of the coefficients, parameter-uniform second order convergenceis obtained [17]. The second order parameter-uniform convergence for the neutron transport equation is due tothe specific scales of its coefficients. For the general form of problem (1.1)-(1.3), we can achieve only first orderparameter-uniform convergence by using piecewise constant approximations of the coefficients. The computedparameter-uniform error parameters using the two-mesh method are given in Table 3.

12

r \ ∆t 1/128 1/256 1/512 1/1024 1/2048

2−1 1.196 ∗ 10−3 5.953 ∗ 10−4 2.970 ∗ 10−4 1.483 ∗ 10−4 7.412 ∗ 10−5

2−2 1.302 ∗ 10−3 6.446 ∗ 10−4 3.207 ∗ 10−4 1.599 ∗ 10−4 7.987 ∗ 10−5

2−3 1.380 ∗ 10−3 6.751 ∗ 10−4 3.340 ∗ 10−4 1.6610 ∗ 10−4 8.283 ∗ 10−5

2−4 1.468 ∗ 10−3 7.024 ∗ 10−4 3.435 ∗ 10−4 1.699 ∗ 10−4 8.449 ∗ 10−5

2−5 1.612 ∗ 10−3 7.411 ∗ 10−4 3.543 ∗ 10−4 1.733 ∗ 10−4 8.568 ∗ 10−5

2−6 1.867 ∗ 10−3 8.104 ∗ 10−4 3.723 ∗ 10−4 1.779 ∗ 10−4 8.700 ∗ 10−5

2−7 2.258 ∗ 10−3 9.366 ∗ 10−4 4.063 ∗ 10−4 1.866 ∗ 10−4 8.917 ∗ 10−5

2−10 2.721 ∗ 10−3 1.361 ∗ 10−3 6.556 ∗ 10−4 2.836 ∗ 10−4 1.174 ∗ 10−4

2−15 2.721 ∗ 10−3 1.364 ∗ 10−3 6.827 ∗ 10−4 3.416 ∗ 10−4 1.708 ∗ 10−4

2−16 2.721 ∗ 10−3 1.364 ∗ 10−3 6.827 ∗ 10−4 3.416 ∗ 10−4 1.708 ∗ 10−4

2−17 2.721 ∗ 10−3 1.364 ∗ 10−3 6.827 ∗ 10−4 3.416 ∗ 10−4 1.708 ∗ 10−4

D∆t 2.721 ∗ 10−3 1.364 ∗ 10−3 6.827 ∗ 10−4 3.416 ∗ 10−4 1.708 ∗ 10−4

p∆t 0.996 0.998 0.999 1.000 p∗ = 0.996

C∆tp∗ 0.685 0.685 0.684 0.682 0.680

Table 5.2: Example 1. Values of D∆tε , D∆t, p∆t and C∆t

p∗ with ε1 = r/16, ε2 = r/4, ε3 = r for various r and ∆t. Wefind p∗ = min∆t p

∆t = 0.996 and C∗p∗ = max∆t C∆tp∗ = 0.685. Here t∗l = tl−1 in (3.3) and (3.4),

Example 2. For a negative εi, a final value must be imposed in order to preserve the maximum principle.In this example, we solve a system with discontinuous coefficients such that for t ∈ [0, 0.5],

−ε1u′1(t) + (5 + e−t)u1(t)− tu2(t)− u3(t)− u4(t) = t,

ε2u′2(t)− u1(t) + (4 + t2)u2(t)− u3(t)− u4(t) = 1,

−ε3u′3 − u1(t)− u2(t) + 5u3(t)− (1 + t)u4(t) = 1 + t,

ε3u′3 − u1(t)− tu2(t)− u3(t) + 5u4(t) = 1− t2,

and for t ∈ [(0.5, 1],

−ε1u′1(t) + (4 + e−t)u1(t)− tu2(t)− u3(t)− u4(t) = 1,

ε2u′2(t)− u1(t) + (4 + t2)u2(t)− u3(t)− u4(t) = 1− t,

−ε3u′3 − u1(t)− u2(t) + (5 + t2)u3(t)− (2 + t)u4(t) = 1− t2,

ε3u′3 − tu1(t)− (1 + t)u2(t)− u3(t) + (4 + e−t)u4(t) = 1 + t,

with the boundary conditions

u1(1) = 0, u2(0) = 0, u3(1) = 0, u4(0) = 0.

This system exhibits initial, final and interface layers, when the |εi| are small. Similarly to [23], let ε1 = r/64,ε2 = r/16, ε3 = r/4, ε4 = r. The numerical solutions of uε1(t) for different r are shown in Figure 5.2. Theresults are consistent with first order parameter-uniform convergence, as expected from the theory.

In Table 5.4, we present the L∞ norm of the numerical errors for different r and time steps, where we haveused t∗l = tl−1 in (3.3) and (3.4). Uniform first order convergence can be observed.

13

r \ ∆t 1/128 1/256 1/512 1/1024 1/2048 order

2−1 1.590 ∗ 10−4 4.011 ∗ 10−5 9.941 ∗ 10−6 2.369 ∗ 10−6 4.738 ∗ 10−7 2.09

2−2 3.047 ∗ 10−4 7.950 ∗ 10−5 1.988 ∗ 10−5 4.748 ∗ 10−6 9.504 ∗ 10−7 2.07

2−3 5.261 ∗ 10−4 1.521 ∗ 10−4 3.934 ∗ 10−5 9.480 ∗ 10−6 1.902 ∗ 10−6 2.02

2−4 7.520 ∗ 10−4 2.624 ∗ 10−4 7.517 ∗ 10−5 1.873 ∗ 10−5 3.792 ∗ 10−6 1.91

2−5 9.316 ∗ 10−4 3.744 ∗ 10−4 1.293 ∗ 10−4 3.570 ∗ 10−5 7.473 ∗ 10−6 1.73

2−6 1.116 ∗ 10−3 4.623 ∗ 10−4 1.835 ∗ 10−4 6.095 ∗ 10−5 1.412 ∗ 10−5 1.55

2−7 1.352 ∗ 10−3 5.512 ∗ 10−4 2.242 ∗ 10−4 8.470 ∗ 10−5 2.342 ∗ 10−5 1.44

2−10 2.502 ∗ 10−3 1.139 ∗ 10−3 4.593 ∗ 10−4 1.449 ∗ 10−4 4.091 ∗ 10−5 1.48

2−15 2.643 ∗ 10−3 1.280 ∗ 10−3 5.978 ∗ 10−4 2.563 ∗ 10−4 8.543 ∗ 10−5 1.22

2−16 2.643 ∗ 10−3 1.280 ∗ 10−3 5.978 ∗ 10−4 2.563 ∗ 10−4 8.543 ∗ 10−5 1.22

2−17 2.643 ∗ 10−3 1.280 ∗ 10−3 5.978 ∗ 10−4 2.563 ∗ 10−4 8.543 ∗ 10−5 1.22

Table 5.3: Example 1. L∞ norm of the numerical errors for different r and time steps, when Al and f l are as in (5.1)and (5.2), and ε1 = r/16, ε2 = r/4, ε3 = r. The last column displays the numerical convergence order by fitting thelog-log plot of the errors.

To look at the uniform convergence order, we present in Table 5.5 the values of D∆tε , D∆t, p∆t and C∆t

p∗ .

Example 3. In this example, we consider the following semi-discretization of the forward-backward parabolicproblem

sign(xi)|xi|p∂ui(t)

∂t=

ui+1(t)− 2ui(t) + ui−1(t)

∆x2− ui(t) + qi(t), 0 < t < T, (5.3)

i = 1, 2, · · · , N − 1,

u0(t) = f(t), uN (t) = g(t), 0 < t < T, (5.4)

ui(1) = γ(x), −1 < xi < 0. ui(0) = s(xi), 0 < xi < 2, (5.5)

with p > 0 andxi = −1 + 3 ∗ i/N 6= 0, i = 1, 2, · · · , N − 1.

In Figure 5.3 we show the numerical results for

T = 0.5, qi(t) = 1, γ(x) = − sin(πx), s(x) = cos2(π

2x), f(t) = 0, g(t) = 1.

Initial and final layers in time occur for both p = 1 and p = 10. The layers become more significant and obviouswhen p is large. We can capture the layers without resolving them (using a lot of nodes in the layer).

6. Conclusion

A tailored finite point method for a multi-scale singularly perturbed system of linear ordinary differentialequations is proposed in this paper. We can give either initial or final values for the ODE system as well as usediscontinuous coefficients.

The tailored finite point method yields approximate solutions that converge in the maximum norm, uniformlywith respect to the singular perturbation parameters. We prove a parameter-uniform error estimate in the

14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

r=1/64

r=1/16

r=1/4

r=1

Fig. 5.2. Example 2. u1(t) for r = 1, 1/4, 1/16, 1/64.

maximum norm and verify our analytical results numerically. To show the performance of our proposed scheme,three numerical examples that exhibit initial and final layers are considered.

In particular, the third example shows the existence of initial and final layers of the solution of the forward-backward parabolic equation. To investigate and understand these layers will be the subject of our futurework.Acknowledgments. H. Han was supported by the NSFC Project No. 10971116. M. Tang is supported byNatural Science Foundation of Shanghai under Grant No. 12ZR1445400.

References

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[2] A. K. Aziz, D. A. French, S. Jensen and R. B. Kellogg, Origins, analysis, numerical analysis, and numericalapproximation of a forward-backward parabolic problem, Modelisation mathmatique et analyse numerique, 33, 5,895-922, 1999.

[3] E.P. Doolan, J. J. H. Miller and W. H. A. Schilders Uniform numerical methods for problems with initial andboundary layers, Boole Press, Dublin, 1980.

[4] P.A. Farrell, A. Hegarty, J. J. H. Miller, E. O’Riordan, G. I. Shishkin, Robust Computational Techniques forBoundary Layers, Applied Mathematics & Mathematical Computation (Eds. R. J. Knops & K. W. Morton),Chapman & Hall/CRC Press. 2000.

[5] H. Han and Z. Huang, A tailored finite point method for the Helmholtz equation with high wave numbers in hetero-geneous medium, J. Comp. Math, 26, 728-739, 2008.

[6] H. Han and Z. Huang, Tailored finite point method for a singular perturbation problem with variable coefficients intwo dimensions, J. Sci. Comp., 41, 200-220, 2009.

[7] H. Han and Z. Huang, Tailored finite point method for steady-state reaction-diffusion equation, Comm. Math. Sci.,8, 887-899, 2010.

[8] H. Han and Z. Huang, An equation decomposition method for the numerical solution of a fourth-order ellipticsingular perturbation problem, Numerical Methods for P. D. E., 28, 942-953, 2012.

[9] H. Han, Z. Huang and B. Kellogg, A tailored finite point method for a singular perturbation problem on an unboundeddomain, J. Sci. Comp. 36, 243-261, 2008.

[10] H. Han, Z. Zhang, Multiscale tailored finite point method for second order elliptic equations with rough or highlyosillatory coefficients, 10, 945-976, 2012.

15

r \ ∆t 1/128 1/256 1/512 1/1024 1/2048

1 1.418 ∗ 10−3 6.920 ∗ 10−4 3.242 ∗ 10−4 1.392 ∗ 10−4 4.645 ∗ 10−5

2−1 1.627 ∗ 10−3 7.813 ∗ 10−4 3.633 ∗ 10−4 1.554 ∗ 10−4 5.176 ∗ 10−5

2−2 1.933 ∗ 10−3 9.222 ∗ 10−4 4.273 ∗ 10−4 1.824 ∗ 10−4 6.070 ∗ 10−5

2−3 2.192 ∗ 10−3 1.036 ∗ 10−3 4.775 ∗ 10−4 2.033 ∗ 10−4 6.756 ∗ 10−5

2−4 2.400 ∗ 10−3 1.118 ∗ 10−3 5.106 ∗ 10−4 2.163 ∗ 10−4 7.165 ∗ 10−5

2−5 2.627 ∗ 10−3 1.194 ∗ 10−3 5.372 ∗ 10−4 2.254 ∗ 10−4 7.428 ∗ 10−5

2−6 2.960 ∗ 10−3 1.293 ∗ 10−3 5.671 ∗ 10−4 2.345 ∗ 10−4 7.658 ∗ 10−5

2−7 3.468 ∗ 10−3 1.451 ∗ 10−3 6.120 ∗ 10−4 2.469 ∗ 10−4 7.949 ∗ 10−5

2−10 4.609 ∗ 10−3 2.166 ∗ 10−3 9.453 ∗ 10−4 3.592 ∗ 10−4 1.056 ∗ 10−4

2−15 4.736 ∗ 10−3 2.293 ∗ 10−3 1.070 ∗ 10−3 4.587 ∗ 10−4 1.529 ∗ 10−4

2−16 4.736 ∗ 10−3 2.293 ∗ 10−3 1.070 ∗ 10−3 4.587 ∗ 10−4 1.529 ∗ 10−4

2−17 4.736 ∗ 10−3 2.293 ∗ 10−4 1.070 ∗ 10−3 4.587 ∗ 10−4 1.529 ∗ 10−4

Table 5.4: Example 2. L∞ norm of the numerical errors for different r and time steps, when t∗l = tl−1 in (3.3) and (3.4),ε1 = r/64, ε2 = r/16, ε3 = r/4, ε4 = r.

[11] P. W. Hemker, A singularly perturbed modal problem for numerical computation, J. Comp. Appl. Math, 76, 277-285,1996.

[12] M. Hochbruck , C. Lubich, On Magnus Integrators for Time-Dependent Schrodinger Equations, SIAM J. NumericalAnalysis, 945-963, 2003.

[13] Z. Huang and X. Yang, Tailored Finite Point Method for First Order Wave Equation, Journal of Scientific Computing49, 351-366, 2011.

[14] H. Han and D. S. Yin, A non-overlap domain decomposition method for the forward-backward heat equation, Journalof Computational and Applied Mathematics, 159, 35-44.

[15] Z. Huang, 2009, Tailored finite point method for the interface problem, Networks and Heterogeneous Media, 4,91-106, 2003.

[16] A. M. Il’in, Difference scheme for a differential equation with a small parameter affecting the highest derivatve,Math. Notes, 6, 596-602, 1969.

[17] S. Jin, M. Tang and H. Han, A uniformly second order numerical method for the one-dimensional discrete transportequation and its diffusion limit with interface, Networks and Heterogeneous Media, 4, 35-65, 2009.

[18] K. Lorenz, T. Jahnke and C. Lubich, Adiabatic integrators for highly oscillatory second-order linear differentialequations with time-varying eigendecomposition, BIT Numerical Mathematics, 45, 91-115, 2005.

[19] C.B. Moler and C.F. Van Loan, Nineteen dubious ways to compute the exponential of a matrix, twenty-five yearslater, SIAM Rev., 45:1, pp. 349, 2003.

[20] J.J.H. Miller, On the convergence ,unifomly in ε, of difference schemes for a two-point boundary singular pertur-bation problem, Numerical analysis of singular perturbation problems, eds, by P.W. Hemker and J.J.H. Miller,Academic Press, 467-474, 1979.

[21] Y. Shih, R.B. Kellogg and Y.Y. Chang, Characteristic tailored finite point method for convection-dominatedconvection-diffusion-reaction problems, Journal of Scientific Computing, 47, 351-366, 2011.

[22] M. Tokman, Efficient integration of large stiff systems of ODEs with exponential propagation iterative (EPI) methods,Journal of Computational Physics, 213, 2, 748-776, 2006.

[23] S. Valarmathi and John J.H. Miller, A parameter-uniform finite difference method for a singularly perturbed lineardynamical system, International Journal of Numerical, Analysis and Modeling, 7, 535-548, 2010.

16

r \ ∆t 1/128 1/256 1/512 1/1024 1/2048

1 7.503 ∗ 10−4 3.678 ∗ 10−4 1.850 ∗ 10−4 9.277 ∗ 10−5 4.645 ∗ 10−5

2−1 8.452 ∗ 10−4 4.180 ∗ 10−4 2.079 ∗ 10−4 1.037 ∗ 10−4 5.176 ∗ 10−5

2−2 1.011 ∗ 10−3 4.950 ∗ 10−4 2.448 ∗ 10−4 1.217 ∗ 10−4 6.070 ∗ 10−5

2−3 1.207 ∗ 10−3 5.588 ∗ 10−4 2.742 ∗ 10−4 1.358 ∗ 10−4 6.756 ∗ 10−5

2−4 1.286 ∗ 10−3 6.140 ∗ 10−4 2.944 ∗ 10−4 1.446 ∗ 10−4 7.165 ∗ 10−5

2−5 1.434 ∗ 10−3 6.569 ∗ 10−4 3.118 ∗ 10−4 1.511 ∗ 10−4 7.428 ∗ 10−5

2−6 1.668 ∗ 10−3 7.256 ∗ 10−4 3.326 ∗ 10−4 1.578 ∗ 10−4 7.658 ∗ 10−5

2−7 2.017 ∗ 10−3 8.394 ∗ 10−4 3.652 ∗ 10−4 1.674 ∗ 10−4 7.949 ∗ 10−5

2−10 2.446 ∗ 10−3 1.221 ∗ 10−3 5.861 ∗ 10−4 2.536 ∗ 10−4 1.056 ∗ 10−4

2−15 2.446 ∗ 10−3 1.223 ∗ 10−3 6.116 ∗ 10−4 3.058 ∗ 10−4 1.529 ∗ 10−4

2−16 2.446 ∗ 10−3 1.223 ∗ 10−3 6.116 ∗ 10−4 3.058 ∗ 10−4 1.529 ∗ 10−4

2−17 2.446 ∗ 10−3 1.223 ∗ 10−3 6.116 ∗ 10−4 3.058 ∗ 10−4 1.529 ∗ 10−4

D∆t 2.446 ∗ 10−3 1.223 ∗ 10−3 6.116 ∗ 10−4 3.058 ∗ 10−4 1.529 ∗ 10−4

p∆t 1.000 1.000 1.000 1.000 p∗ = 1.000

C∆tp∗ 0.626 0.626 0.626 0.626 0.626

Table 5.5: Example 2. Values of Dε, D, p, p, C∗p and C∗p with ε1 = r/64, ε2 = r/16, ε3 = r/4, ε4 = r, for various r and∆t. We find p∗ = min∆t p

∆t = 1.000 and C∗p∗ = max∆t C∆tp∗ = 0.626. Here t∗l = tl−1 in (3.3) and (3.4),

17

a)

−1

−0.5

0

0.5

1

1.5

2

0

0.1

0.2

0.3

0.4

0.5

0

0.2

0.4

0.6

0.8

1

xt 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

t

u

b)

−1

−0.5

0

0.5

1

1.5

2

0

0.1

0.2

0.3

0.4

0.5

0

0.2

0.4

0.6

0.8

1

xt0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

t

u

Fig. 5.3. Example 3. The numerical solution of the forward-backward parabolic problem. Left: ui(t) for i = 1, 2, · · · , N ;Right: uM where M satisfies xM < 0 and xM+1 > 0. a) p = 1; b) p = 10. Here ∆x = 1/40, ∆t = 1/256.